On the convergence of the Newton-GMBACK method


When the GMBACK solver is used in the Newton iterates, the iterates may be written either as inexact Newton iterates or as quasi-Newton iterates. In this paper present results which assure the local convergence of the iterates in the both settings.


Emil Catinas
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy


linear systems; Krylov solvers; GMBACK; backward error.

Cite this paper as:

E. Catinas, On the convergence of the Newton-GMBACK method. 2007 International Conference on Engineering and Mathematics, Bilbabo, July 9-11, 2007, pp.11-14.



Not available yet.

About this paper

Publisher Name
Print ISSN
Online ISSN





[1 E. Catinas,  Inexact perturbed Newton methods and applications to a class of Krylov solvers,  I. Optim. Theory, Appl. 108 (2001), 543-570.

[2] E. Catinas,  On the superlinear convergence of the successive approximations method,  J. Optim. Theory Appl., 113 (2002), 473-485.

[3] E. Catinas,  The inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005), no. 249, pp. 291-301.

[4] R. S. Dembo, S.C. Eisenstat and T. Steihaug,  Inexact Newton methods, SIAM, J. Numer. Anal. 19 (1982), 400-408.

[5] J. E. Dennis, Jr. and J. J. More,  A characterization of superlinear convergence and its application to quasi-Newton methods,  Math. Comp. 28 (1974), 549-560.

[6] E. M. Kasenally, GMBACK: a generalised minimum backward error algorithm for nonsymmetric linear system, SIAM J. Sci. Comput. 16 (1995), 698-719.

[7] E. M. Kasenally and V. Simoncini,  Analysis of a minimum perturbation algorithm for nonsymmetric linear systems, SIAM J. Numer. Anal. 34 (1997), 48-66.

[8] B. Morini, Convergence behaviour of inexact Newton methods, Math. Comp. 68 (1999), 1605-1613.

[9] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[10] F. A. Potra, On Q-order and R-order of convergence, J. Optim. Theory Appl. 63 (1989), 415-431.



Related Posts